Since strict quasiconcavity implies diminishing marginal rate of substitution. Robinson 1933 in her comparably classic book on the economics of imperfect competition. We quantify the size of strict improvements that imposing shape restrictions bring. Necessary and sufficient conditions for a maximum 10 5. In general, a strictly concave funcon must bestrictlyquasiconcave, although the converse is not true. Abstract concavity of increasing coradiant and quasi. Concavity and quasiconcavity in economics adriani, veronica a. John riley minor corrections 25 july 2016 concave functions in economics 1. Roughly speaking concavity of a function means that the tangent is above graph. You can see some convexity and concavity of functions,mathematical methods in economics2 sample questions with examples at the bottom of this page. Concavity, second order conditions, and so on october 5, 2005. Concavity and quasiconcavity 601 if strict inequality holds, namely fx fv, then we can find a point p on the line segment vy such that fx fp and derive a contradiction as before. This is having a y set that is strictly convex along the same as having strictly convex its upper boundary. In this paper, we first prove the strict quasiconcavity of maximizing functions, and next, using a generalization of the kkm theorem, we prove two maximum theorems without assuming the upper.
These notes are the starting point for a mathpreparation book, primarily for use by ucsd students enrolled in econ 205 potentially for use by folks outside ucsd as well. The twist is that while concavity requires that v0d2fxvbe negative for all v, quasiconcavity requires only that v0d2fxvbe negative for vsuch that rfx v 0. Concavity and quasiconcavity in economics sciencedirect. Fundamental preference axioms completeness transitivity continuity greed strict quasiconcavity smoothness. The twist is that while concavity requires that v0d2fxvbe negative for all v, quasi concavity requires only that v0d2fxvbe negative for vsuch that rfx v 0. In this paper, we study nonnegative, increasing, coradiant and quasiconcave functions over real locally convex topological vector spaces. This strict version of quasiconcave differences thus parallels strict quasiconcavity in securing a unique optimizerbut as was just explained the conditions are logically independent of each other neither implies the other. In the rush of rstsemester graduate economics, every economist encounters con. Inferior goods, monotonic utility and strict concavity. The lay reader should also be warned that quasi rational economics is, at beast, only quasi accessible. Whats the intuitive difference between quasi concavity and concavity. Convex analysis is the standard tool for analyzing textbook economics. Property, which is always satisfied in the particular case where h is a bilinear form, is nothing but the requirement on the function h of quasi concavity in the variable x and of quasi convexity in the variable y. X the notbetterthanx set and the notworsethanx set are closed in x.
Some level curves of two functions that are not strictly quasiconcave, although. In this paper, we first prove the strict quasi concavity of maximizing functions, and next, using a generalization of the kkm theorem, we prove two maximum theorems without assuming the upper. A function with the property that for every value of a the set of points x, y such that f. In this paper, we study nonnegative, increasing, coradiant and quasi concave functions over real locally convex topological vector spaces. Thaler was awarded the 2017 nobel prize in economics. If you want to change your field in more fundamental ways, then obtain your primary motivation from life, and use it to look for fundamental shortcomings of previous thinking in the field. Find all the books, read about the author, and more. If we model the surface of the mountain as a function f of its longitude and latitude x, y, then a contour is a level curve of f. Social equilibrium theorems with strict quasiconcavity. Quasiconcave utility functions get that name because quasiconcavity is. As such, this volume is more of an anthology than a coherent wholeand reads like it. Math prep notes uc san diego department of economics.
But we have gone through the theory of the firm first for a good reason. Now, if we assume the additional assumption on the constraint multimap c i in theorem 3. Eric rasmusen, department of business economics and public policy. Such functions have frequently been employed in microeconomic analysis. Research methodology in applied economics don ethridge. However, h f bracketleftbigg 0 2 x 2 x 2 bracketrightbigg so.
Inequality and quasiconcavity by kristof bosmans center for. This volume collects many of his early studies of behavior that departs from the predictions of rational choice economics. Characterizing uncertainty aversion through preference for. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. You can see some convexity and concavity of functions,mathematical methods in economics 2 sample questions with examples at the bottom of this page.
Notes on microeconomic theory these are summaries from. Maximum theorems with strict quasiconcavity and applications. A theory of uncertainty aversion may be characterized by the pairs of acts for which strict preference for a mixture between them is permitted. In mathematics, a quasiconvex function is a realvalued function defined on an interval or on a. That is, whether or not a function is concave depends on the numbers which the function assigns to its level curves, not just to their shape. Let a twice continuous differentiable, c2 function f defined on some open domain n t 22be given, and. We will say that the solution is interior if x i0, 8i. New maximum theorems with strict quasiconcavity request pdf. Pigou 1934 also contributed to the understanding of the concept at its genesis. Quasiconcavity is a property of functions which, if strict, guarantees that a. Inferior goods, monotonic utility and strict concavity duplicate ask question asked 3 years. Dynamic economics is about explaining economic behaviors through time.
Quasiconcavity is a property of functions which, if strict, guarantees. It is used to show the important result that for a concave differentiable function f every point x for which f x 0 is a global maximizer, and for a convex differentiable function every such point is a global minimizer. Metric spaces a linear or vector space x is a set of elements together with addition and scalar multiplication operators. It is frequently plausible in economic applications that a function fx being max. The origins of the notion of quasiconcave function are considered, with special. Abstract concavity of increasing coradiant and quasiconcave. Can you give an example of a quasiconcave function that is not concave. Economics convexity and concavity of functions,mathematical methods in economics2 summary and exercise are very important for perfect preparation. This book is quite comprehensive, but not quite readable. Textbook expositions of quasiconcavity can be found in kreps 1990 p. Convex preferences get that name because they make upper contour sets convex. In many such cases, one nds it easier to defend the assumption that these functions are quasiconcave. A strictly quasiconvex function has strictly convex lower contour sets, while a.
Whats the difference between quasiconcavity and concavity. If you graph a mathematical function and the graph looks more or less like a badly made bowl with a few bumps in it but still has a depression in the center and two ends that tilt upward, that is a quasiconcave function. R is concave convex if and only if its restriction to every line segment of rn is concave convex function of one variable. Quasiconcave is a topological property that includes concavity. To understand the significance of the terms applications in economics, it is useful to begin with a brief consideration of the origins and meaning of the term in mathematics. A concave function that has no linear parts is said to be strictly concave. Concavity and quasiconcavity in economics luissthesis. In the arrowdebreu model of general economic equilibrium, agents have convex budget sets and convex preferences. Arbitrage pricing arbitrage profit average cost balance of payments budget constraint call option concave function consumer surplus consumption function convex function deadweight loss demand curve econometrics economic agent economic model economics economics textbook elasticity endogenous endogenous technical change. Owing to strict quasiconcavity of f, tis strictly monotonic, hence invertible. We next characterize the abstract concavity, the upper support set and the superdifferential of this class of functions by applying a type of duality, which is used in microeconomic. The next result states this observation, and the similar one for convex functions, precisely.
In all answers where you provide a counterexample, you must show that your example is really a counterexample. Lecture notes on elasticity of substitution ted bergstrom, ucsb economics 210a march 3, 2011 todays featured guest is \the elasticity of substitution. I wonder how i can show that this production function in capital k and labour l is quasi concave but not strict concave. Let a twice continuous differentiable, c2 function f defined on some open domain n t 22be given, and let xt, let ax t, tn 1,, be the 11tt u x x x xw w w w. This also means that if a monotonic transformation of f is concave, then f is concave. The proof of strict quasiconcavity follows from a generalisation of minkowskis inequality proved in hardy et al. Property, which is always satisfied in the particular case where h is a bilinear form, is nothing but the requirement on the function h of quasiconcavity in the variable x and of quasiconvexity in the variable y. Concavity is easier to understand than quasiconcavity, and concave functions on compact sets also have a single, global maximum, but concavity is a much stronger assumption. The lay reader should also be warned that quasi rational economics is, at beast, only quasiaccessible. Elasticity of a function of a single variable before we meet this guest, let us spend a bit of time with a slightly simpler notion, the elasticity of a a function of a single variable. Roughly speaking concavity of a function means that the graph is above chord. My underlying idea is that whit an inferior good, there exist a point where the utility consuming less of one good is better and then its derivative negative at that point, moreove, i think that this behaviour would break the concavity shape. This paper provides such a characterization for two leading representations of uncertainty averse preferences. The most important is probably that strict concavity implies that consumption growth depends on the level of wealth, which is serially correlated, so the implication of hall 1978 that consumption should follow a random walk at the household level no longer.
Can you give an example of a quasi concave function that is not concave. Quasiconcave utility functions get that name because quasiconcavity is a weaker property than concavity. Some treatments of microeconomics handle consumer analysis first. Tesi di laurea in mathematics, luiss guido carli, relatore marco dallaglio, pp. I wonder how i can show that this production function in capital k and labour l is quasiconcave but not strict concave. When i say intuition i mean what property of the strategic interaction in the game is captured by the strict concavity condition. Inequality and quasiconcavity by kristof bosmans center. Social equilibrium theorems with strict quasi concavity. Fromnow on we will assume thatx is aconvex subset of rn. Concave andquasiconcave functions 1 concaveandconvexfunctions 1. It is with these economic applications in mind that arrow and enthoven developed the theory of quasiconcave programming.
Quasiconcave, but not strictly quasiconcave and not strictly concave. We shall demonstrate thjs more formally in the paragraphs that follow. Quasiconvexity and quasiconcavity extend to functions with multiple. In mathematics, a quasiconvex function is a realvalued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form. Quasi concavity again we begin with some geometrically suggestive definitions. One problem with concavity and convexity which well encounter again when we look at homogeneity is that they are cardinal properties. Other results of this work are a strict converse duality theorem for mathematical programming. I if f is a monotonic transformation of a concave function, it is quasiconcave. John riley minor corrections 25 july 2016 ucla econ. Some treatments of micro economics handle consumer analysis first. Apr 10, 2019 quasiconcave is a mathematical concept that has several applications in economics. On the origin of rconcavity and related concepts econstor.
Concavifying the quasiconcave kelley school of business. This theorem is the theoretical ground behind the mechanical di. I if f is concave, then it is quasiconcave, so you might start by checking for concavity. Uncertainty aversion is often modelled as strict quasiconcavity of preferences over uncertain acts. We next characterize the abstract concavity, the upper support set and the superdifferential of this class of functions by applying a type of duality, which is used in. Quasiconvex functions have applications in mathematical analysis, in mathematical optimization, and in game theory and economics. Quasiconcave is a mathematical concept that has several applications in economics. At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. Economics stack exchange is a question and answer site for those who study, teach, research and apply economics and econometrics. Economics 210a october 2011 the exam has 6 questions. The profit function is the convex conjugate of the cost function. What you write is essentially one of the results in the original rosens paper. These notes are the starting point for a mathpreparation book, primarily for use by.
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